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arXiv:2603.27894

Quantum Robust Control using Geometric Optimal Control Theory

Francesca Albertini, Domenico D'Alessandro

quant-phSubmitted Mar 29, 2026• Announced Mar 31, 2026• 0 citations
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Abstract

In this paper, we demonstrate an approach to quantum robust control based on the tools of geometric optimal control. The central objects of interest are the sensitivity functions defined as the coefficients in the Taylor expansion of the trajectory with respect to the (unknown, small) parameters which describe the deviation of the actual model from nominal one. In terms of these quantities, we formalize an optimal control problem where one searches for the optimal nominal trajectory which minimizes the size of the sensitivity while taking into account other aspects of the control design such as the energy of the control field. We consider in detail the case of a single qubit with a dephasing Hamiltonian term, and the optimal control problem of obtaining a state transfer by minimizing the weighted sum of the energy of the controlling field and the first order sensitivity. At the limit of a very large weight on the sensitivity, we obtain the optimal control which zeros the sensitivity and minimizes the control field energy. This problem has a rich mathematical structure which enables its solution in terms of elliptic integrals. For this problem, we obtain an explicit solution which is particularly simple and also smooth, avoiding discontinuities which are present in other approaches. We extend the results to the robust control of two quantum bits minimizing cross-talk contamination, as we show that such a problem decouples in two independent one qubit problems.

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In this paper, we demonstrate an approach to quantum robust control based on the tools of geometric optimal control. The central objects of interest are the sensitivity functions defined as the coefficients in the Taylor expansion of the trajectory with respect to the (unknown, small) parameters which describe the deviation of the actual model from nominal one. In terms of these quantities, we formalize an optimal control problem where one searches for the optimal nominal trajectory which minimizes the size of the sensitivity while taking into account other aspects of the control design such as the energy of the control field.
We consider in detail the case of a single qubit with a dephasing Hamiltonian term, and the optimal control problem of obtaining a state transfer by minimizing the weighted sum of the energy of the controlling field and the first order sensitivity. At the limit of a very large weight on the sensitivity, we obtain the optimal control which zeros the sensitivity and minimizes the control field energy. This problem has a rich mathematical structure which enables its solution in terms of elliptic integrals. For this problem, we obtain an explicit solution which is particularly simple and also smooth, avoiding discontinuities which are present in other approaches.
We extend the results to the robust control of two quantum bits minimizing cross-talk contamination, as we show that such a problem decouples in two independent one qubit problems.
The design of optimal final-state controllers of quantum-mechanical systems that are insensitive to errors in the molecular Hamiltonian or to errors in the initial state of the system is considered. Control arises through the interaction of the system with an external field; the goal is optimal design of these latter fields for various physical objectives in the presence of system uncertainty. Sensitivity to modeling errors and other uncertainties in the molecular Hamiltonian is minimized by considering averaged costs for a family of Hamiltonian functions H0{\mathit{H}}_{0}H0​(\ensuremath{\alpha}) indexed by the random variable \ensuremath{\alpha} taking values on a compact set in Euclidean space. Similarly, sensitivity of the optimal control to the initial state is minimized by viewing the initial condition as a Hilbert-space-valued random variable and considering an optimization problem with a cost functional that is averaged over the class of initial conditions. A precise formulation of the control problem is given, and its well-posedness is established. Cost propagators are defined to display the dependence of the performance index on the initial conditions explicitly, which allows analytic averaging of initial conditions.The constrained optimization problem is reduced to an unconstrained optimization problem by the introduction of Lagrange-multiplier operators. Necessary conditions for the unconstrained problem provide the basis for a gradient search for an optimal solution. Finite-difference schemes are utilized to provide a numerical approximation of the optimal control problem. Numerical examples are given for final-state control of a diatomic molecule represented by a Morse potential illustrating design for systems with initial-phase uncertainty and parametric uncertainty. The resultant insensitive controllers execute different strategies depending on the design requirements. The controller designed to be insensitive to errors in the initial phases adopts a strategy of phase imprinting during the initial stages of the control interval to compensate for a lack of knowledge of the initial phases. It is also shown that it is not possible to coerce a system from a state with completely random initial phases to a correlated state using the class of averaged controllers considered here. The controller designed to be insensitive to parametric Hamiltonian errors adopts a strategy of amplitude restraint to prevent the wave packet from taking significant excursions into the regions where the potential is uncertain. The interesting structure exhibited by the controllers in response to the different design requirements, and the superior performance of the insensitive controllers when compared with controllers designed at nominal phases and parameters, illustrate the usefulness of the cost-averaging technique for design in the presence of uncertainties.
We develop an inverse geometric optimization technique that allows the derivation of optimal and robust exact solutions of low-dimension quantum control problems driven by external fields. We determine in the dynamical variable space optimal trajectories constrained to robust solutions by Euler-Lagrange optimization; the control fields are then derived from the obtained robust geodesics and the inverted dynamical equations. We apply this method, referred to as robust inverse optimization (RIO), to design optimal control fields producing a complete or half population transfer and a not quantum gate robust with respect to the pulse inhomogeneities. The method is versatile and can be applied to numerous quantum control problems, e.g., other gates, other types of imperfections, Raman processes, or dynamical decoupling of undesirable effects.
In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.
Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of un-desired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.
The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time-dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently. The accurate numerical computation of optimal controls of infinite dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2 versus H1 based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques. A matrix-free cascadic BFGS algorithm is introduced in the L 2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented. A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov-Newton scheme applied to a norm-preserving discrete model of a dipole quantum control problem. The use of second-order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first- and second-order optimality conditions. Criteria for the discretization of the non-convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust linesearch procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high-energy state transitions. Control of bound-to-bound and bound-to-continuum transitions in open quantum systems and vector field control of two-dimensional systems is presented. An efficient space-time spectral discretization of the time-dependent Schrodinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined.
Robust quantum control usually assumes one of two extremes: the error is of a known form and only its strength is unknown, or nothing at all is known about it. In practice one often sits in between. The size of a static error is unknown, but its class is known, for example, it only shifts energy levels, or it acts on each subsystem separately. We ask how fast a control can cancel, to first order, every error in such a class. A time-independent Hamiltonian can do this if and only if all errors in the class have zero diagonal in a common basis. In that case an equally spaced energy ladder in this basis does it after one period. A lower bound that holds for any time-dependent control with bounded drive power shows that this simple constant control is optimal for several natural classes, including unknown level shifts and unknown local errors on two subsystems. Completely unknown errors need a time-dependent control, and there, knowing the error class shortens the minimum time by a factor d2+1\sqrt{d^2+1}d2+1​. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.